it actually depends on the filter. the filter that my roomate brought makes really awful-tasting water (confirmed by several people who have tasted it thus far), I think it depends on the filter membrane or whatever that the filter is lined with ... i think this one is carbon-based or something so it gives a really weird flavour.
One time on TV I saw a test about bottled water. They gave 1000 people 6 samples of water.
One was bottled water, 4 were tap water, one was distilled urine. There was not a single person who could genuinely tell the difference between any of them (because there is none), obviously, 1/6th of people correctly guessed the bottled water, but that is to be expected and even if they did not try the samples the results would have been similar.
The test then went on to produce and sell bottled water, it was relatively cheap compared to standard bottled water... Oh yeah, the water used was tap water. On the trials they did with the product, not a single person could tell was just tap water in the bottle.
I honestly cannot believe that the tap water from where ever you live tastes to bad you would resort to spending several thousands of times more on bottled water. I am guessing most of the "bad tasting" water is more psychological than anything, similar to a placebo effect. Because you know it is tap water, you are basically tricking yourself into thinking it is worse than it really is.
People who buy and drink only bottled water are pretty crazy if you ask me!